From Distinguishability to History Cascades: Projections, Memory, and Stable Transport

We develop a self-contained geometry of averaging shared by changes of resolution and the evolution of history. On a finite weighted carrier, preservation of the mass of every block uniquely determines conditional averaging; the resulting operator is a positive orthogonal projection. Nested resolutions split fields into orthogonal detail bands and yield exact Pythagorean identities, including the full Hermitian Gram matrices of complex-valued families. A single binary readout transported along a reversible finite history generates an increasing cascade of memory windows. Successive projection increments are orthogonal, step entropy decreases with window length and telescopes to the information added by the complete history, while directed transition counts separate into a self-adjoint component and an anti-Hermitian component carrying orientation. Counts, compressions, and their Dirichlet forms aggregate exactly between adjacent windows, making refinement and coarse-graining two directions of the same orthogonal geometry. We then study dynamics and imperfect transport. Projection memory, Schur complementation, and moving resolutions arise from the same visible–hidden decomposition. Local compatibility defects telescope to the error of the complete transported history. Sharp margin estimates separate the irreducible loss caused by the initially hidden sector from the additional cost accumulated by sequential coarse transport. Under summable defects and transverse mixing, the transported history converges in operator norm. Under defects summable in Hilbert–Schmidt norm and isometric transport, it admits an exact defect-free shadow with an explicit correction cost and control of every finite full Gram. The resulting calculus connects distinguishability, multiscale averaging, memory, and stable history transport within one verifiable construction.